Before the advent of geared drivetrains for bicycles, the pedals and cranks were directly connected to the hub.  On these “direct-drive” bikes, one rotation of the pedals corresponded with one rotation of the drive wheel.  Accordingly, wheel size and pedaling rate alone determined a rider’s speed.  If a standard modern road bike used a direct-drive transmission, a rider pedaling at 80 revolutions per minute would travel at a speed of just 6 miles per hour. With virtually all cyclists pedaling under 100 revolutions per minute for any sustained period, the bikes required comically large wheels to achieve a reasonable speed.

A direct-drive Penny-Farthing bike. Photo Credit: http://upload.wikimedia.org/wikipedia/commons/9/99/Velocipedist.JPG

At the same cadence of 80 revolutions per minute, a direct-drive “penny-farthing” bike like the one shown above with a 5ft tall drive wheel would cruise at 14 miles per hour.  With a very high center of mass located just behind the front hub, riders often went flying head first to the ground upon sudden deceleration on these dangerous machines. It’s no surprise that direct-drive bicycles were quickly supplanted by safer, more practical geared bikes.  In this post I’ll explore the physics of gears and the elements of speed on a bike.

## Gears and Gear Ratios Explained

Geared bicycle drivetrains allow for one revolution of the cranks to generate far more and, if needed, fewer than one rotation of the drive wheel.  The ratio of the rotational rate of the drive wheel to that of the cranks is known as the gear ratio of a bicycle. For example, a gear ratio of 3 means that one rotation of the cranks corresponds to three rotations of the drive wheel.

On modern multi-geared bicycles, the cranks are connected to a single, pair, or triplet of chainrings and the drive wheel hub is connected to a cassette consisting several sprockets. A chain connects the chainrings to the cassette sprockets, transmitting the power generated at the cranks to the drive wheel.  A front and rear derailleur allow the rider to change between different size chainrings and cassette sprockets.

Let’s suppose the chain links a 44-tooth chainring to an 11-tooth sprocket.  One rotation of the crankset rotates the chainring once, which translates the chain forward by 44 links.  This translation turns the sprocket (and the drive wheel) by:

$\dfrac{44\:links}{11\:\frac{links}{rotation}} = 4\: rotations$

Thus the gear ratio is equal to:

$\dfrac{chainring\:teeth}{sprocket\:teeth}$

Riding in the smallest chainring and largest sprocket minimizes the gear ratio. Conversely, riding in the largest chainring and smallest sprocket maximizes the gear ratio.

## A formula for Speed

For every full rotation of the drive wheel of diameter d meters, a bike moves forward by a distance π·d meters, the circumference of the drive wheel.  Therefore, as long as pedaling continues to speed up a bicycle, the speed of the bike with a wheel diameter d, pedaling cadence c, and gear ratio g is given by:

$speed = {c\:[\frac{crank\,revolutions}{minute} ]\cdot g\: [\frac{wheel\,revolutions}{crank\,revolution}] \cdot \pi d\: [\frac{meters}{wheel\,revolution}]}= {\pi c g d \:[\frac{meters}{minute}]}$

The speed in miles per hour is given by:

$\pi c g d \:[\frac{meters}{minute}] \cdot [\frac{1\:mile}{1609\:meters} ]\cdot [\frac{60\:minutes}{hour}] = \dfrac{60 \pi c g d}{1609}\: [\frac{miles}{hour}]$

For example, a rider pedaling a standard¹ road bike with a cadence of  80 rpm with a gear ratio of 3.1 will move at a speed of:

$60 \pi \cdot 80 [\frac{crank\: rev}{minute}] \cdot 3.1 [\frac{wheel\:rev}{crank\:rev}] \cdot 0.672 [\frac{meters}{wheel\:rev}]\cdot [\frac {1\:mile}{1609\:meters}] = 19.5 [\frac{miles}{hour}]$

## Finding the Perfect Gear Range

A key component to fast and efficient riding is having the appropriate gear range for your fitness level and riding conditions. Your lowest gear should allow you to maintain a high cadence on the steepest climbs you ride, and your highest gear should allow you to continue to build speed on long downhills instead of “running out of gears.”

Gearing choices are essential to smooth riding on hilly courses

Suppose you want to be able to pedal at 80 rpm while traveling at your current maximum speed of 8 mph on the steepest hills you climb, and that you want to be able reach 45 mph when spinning at 120 rpm in your biggest gear on downhills.  Setting v = 8 mph, c = 80 rpm, d = 0.672 meters (for a standard¹ road bike), and solving for g gives 1.27: the gear ratio needed to maintain a cadence of 80 rpm while pedaling uphill on your steepest climbs. Setting c = 120 rpm, v = 45 mph, and r = 0.672 m and solving for g gives: 4.76.  Thus an ideal transmission in this scenario has a gear ratio range of 1.27 to 4.76.

A standard road transmission has 39 and 53 tooth chainrings and a sprocket range of 11-28 teeth.  The smallest gear ratio = smallest chainring / largest sprocket = 39/28 = 1.39, and the largest gear ratio = largest chainring / smallest sprocket = 53/11 = 4.82. Using our equation for c,v,g, and r, the rider would have to settle for a cadence of 73 rpm at 8 mph on the steepest climbs, or increase his fitness to hit 8.75 mph at 80 rpm. Spinning in the 53-11 combination at 120 rpm however, the rider will reach a speed of 45.5 mph.

Accordingly, if you average much less than 8 mph on your steepest climbs, you may want to consider a “compact” 50/34 chainring combination and/or switching to a wider 11-32 cassette. Also, if you have no interest in pedaling faster than 35 mph, the smaller gear ratio jumps in a 50/34 chainring set will help you find the right gear for the terrain.

This knowledge is utilized by bike engineers who design drivetrain systems, professional bike racers and mechanics who need to select the best system for the race at hand, and by enthusiastic amateurs who leverage their knowledge of physics to optimize their riding experience.

In a future Powered by Physics post, I’ll dig deeper into the pros and cons of different road and mountain drivetrain systems.

Notes:
1. A standard 700C road bike has a bead seat diameter of 622 mm and 25mm tires. Thus, the diameter of the wheel is roughly the full bead seat diameter plus twice the height of the tires: 0.622 meters + 2·0.025 meters = 0.672 meters.